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Article Contents

# Topological stability in set-valued dynamics

• * Corresponding author: Carlos Arnoldo Morales Rojas
Work partially supported by CNPq from Brazil and MATHAMSUD 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.
• We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

Mathematics Subject Classification: Primary:54H20;Secondary:54C60.

 Citation:

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